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A Historical Review of the Classifications of Lie Algebras

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A Historical Review of the Classifications of Lie Algebras REVISTA DE LA UNION´ MATEMATICA´ ARGENTINA Vol. 54, No. 2, 2013, Pages 75{99 Published online: December 11, 2013 A HISTORICAL REVIEW OF THE CLASSIFICATIONS OF LIE ALGEBRAS LUIS BOZA, EUGENIO M. FEDRIANI, JUAN NU´ NEZ,~ AND ANGEL´ F. TENORIO Abstract. The problem of Lie algebras' classification, in their different vari- eties, has been dealt with by theory researchers since the early 20th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimen- sion. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of alge- bras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research. 1. Introduction About 1870, the Norwegian mathematician Sophus Marius Lie (1842{1899) be- gan to study some types of geometric transformations that promised to have con- siderable relevance in the subsequent study of symmetries. No one could have imagined at that moment the impact of his discoveries: the gestation of key tools for the development of Modern Physics and, in particular, the Theory of Relativity. We can currently state that Lie Theory, regarding Lie groups as well as Lie alge- bras, has proved to be the key to solving many problems related to Geometry and to Differential Equations, which links theoretical Mathematics to the real world. Owing precisely to their tangible applications, scientists of different disciplines have used specific examples of Lie algebras over different fields and in different dimensions, everyone according to their needs. However, mathematicians are usu- ally more interested in generality than in obtaining a few examples. That is why it was reasonable to expect the immediate interest aroused in the mathematical community by the apparent complexity and the amazing elegance of the new al- gebraic structures. As shown throughout this document, hundreds of publications attest the attempts made so far to categorize these objects. However, not only has a general classification of Lie algebras not been reached, but the objective has been achieved in very few families of algebras for any dimension and field. The different strategies followed by each author have fallen by the wayside, as well as 2010 Mathematics Subject Classification. 17B; 17-01; 17-02; 17-03; 17B99. Key words and phrases. Lie algebra, classification, current status. 75 76 L. BOZA, E. M. FEDRIANI, J. NU´ NEZ,~ AND A.´ F. TENORIO partial classifications given by each research group, some of them wrong and cor- rected in subsequent articles, or discoveries used to make classification easier. We especially point out here the search for invariants, in most cases designed to enable a specific classification but proved useful later for other reasons, which provided a real advance of scientific knowledge. Many studies have been written, in the field of Lie algebras, on analysis or representation of properties. However, specific attention will be given in this article to the issue of classifications because it would be impossible to collect in a reasonable space all the publications related to Lie Theory. Even if we limit ourselves to classifications, in order to enable the subsequent presentation of the different research lines and the achievements so far, we must first clarify some key concepts. The first one is actually that of classification. In Mathematics, to classify is to find a certain property allowing the definition of an equivalence relation among the elements of a set so that this set will be divided into disjoint classes. However, it is interesting that on finishing a classification we are able to choose an element as representative of each equivalence class. Furthermore, in the ideal situation it would also be useful to be able to assign in a simple way each element to its corresponding class. Regarding the elements we want to classify, we must not forget that every Lie algebra is a vector space upon which a new binary operator is defined. This means that in each case a field must be set over which a vector space is defined and, also, that a particular dimension can be chosen for the vector space. In the case of Lie algebras, for the reasons already suggested, additional properties distinguishing be- tween those algebras that verify them and those which do not are often considered. Every time such a property is described there is a chance to classify the Lie algebras that verify it, also in every field and dimension, as happened in the general case. Sometimes, these properties are also useful to classify wide families of algebras; this fact supports their importance. However, the classifications generated from the consideration of different properties are not compatible in general. This fact makes the following description difficult and intricate sometimes. Finally, we must make another short comment about the techniques used to classify. Although it is not the purpose of this work to describe all the possi- ble strategies for classifying Lie algebras, we consider it essential to recognize the role played by computing in this field. Of course, the first classifications were not made with the help of computers, what considerably reduced the mathematicians' chances and significantly increased the likelihood of making mistakes. Taking into account that the increase in dimension involves a considerably larger increment in the complexity of the operations performed, it is logical to admit that com- puting power, as well as the development of increasingly versatile and effective symbolic computation packages, make possible studies that were unfeasible years ago. Therefore, on these and other grounds, progress in the classifications of Lie algebras is not only determined by the consideration of new families of algebras but also because over the course of time the same problems can be tackled but in upper dimensions. Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013) A HISTORICAL REVIEW OF THE CLASSIFICATIONS OF LIE ALGEBRAS 77 This article consists of four parts, apart from this introduction. The primary classification development is described in the next section. Afterwards, we describe the partial classifications comprising subsets of the whole collection of algebras to be studied. Such works are completed with other classifications partially related to the traditional Lie algebras. Finally, we present some brief concluding remarks. 2. Heart of the classifications Classifying all Lie algebras of dimension less than 4 is an elementary exercise. However, when considering dimension 4, complete classifications are much harder, and subsequent classifications usually refer to subclasses. As it is already well known, there exist three different types of Lie algebras: the semi-simple, the solv- able, and those which are neither semi-simple nor solvable. So, determining the classification of Lie algebras, in general, is equivalent to revealing the classification of each of these three types. However, by the Levi{Maltsev Theorem, which is the combination of the results formulated firstly by Levi [98] in 1905, and later by Malt- sev [107] (note that it can also be written Malcev) in 1945: any finite-dimensional Lie algebra over a field of characteristic zero can be expressed as a semidirect sum (the Levi-Maltsev decomposition) of a semi-simple subalgebra (called the Levi fac- tor) and its radical (its maximal solvable ideal). It reduces the task of classifying all Lie algebras to obtaining the classification of semi-simple and of solvable Lie algebras. Nevertheless, as we will see later, this procedure is not entirely valid when dealing with fields of a given positive characteristic. Let us consider first the standard case, over the complex and the real fields. With respect to the first problem, the classification of semi-simple Lie algebras was completely solved by the well-known Cartan Theorem: any semi-simple complex or real Lie algebra can be decomposed into a direct sum of ideals which are simple subalgebras being mutually orthogonal with respect to the Cartan-Killing form. So, the problem of classifying semi-simple Lie algebras is then equivalent to that of classifying all non-isomorphic simple Lie algebras; and the classification of sim- ple Lie algebras was already obtained by Killing, Cartan, and others in the last decade of the 19th century [44]. Hence, it can be admitted that the problem of the classification of semi-simple Lie algebras is at present totally solved. Indeed, mainly Killing and Cartan, although other authors also worked in this subject, classified simple Lie algebras in five different classes (the so-called simple classical Lie algebras): the algebras belonging to the linear special group, those odd orthog- onal algebras, the even orthogonal algebras, the symplectic algebras, plus five Lie algebras having no relation among them and not belonging to any of the previous classes, which were called by the authors exceptional or exotic Lie algebras. With respect to the classification of solvable Lie algebras, in spite of the first attempts by Lie [99, 100] and Bianchi [27], it can be said that Dozias was one of the first authors who faced that problem seriously, in 1963: he classified in his Ph.D. thesis the solvable Lie algebras of dimensions less than 6 over the field of the real numbers [57]. In this same year Mubarakzjanov (see [111, 112, 113] and [119], too) also classified these algebras up to dimension 6 over the field of real numbers. Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013) 78 L. BOZA, E. M. FEDRIANI, J. NU´ NEZ,~ AND A.´ F. TENORIO Some decades later, in 1990, Patera and Zassenhaus obtained the classification of solvable Lie algebras up to dimension 4 over any perfect field [120], and in that same year Turkowski [161] dealt with solvable Lie algebras of dimension 6, includ- ing the classification of the Levi decomposable algebras, which are semidirect sum of semisimple and solvable Lie algebras (see [160, 162]).
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